Sustainable Energy - Without the Hot Air

4.1. Cars II http://www.ck12.org

Figure A.2:Our cartoon: a car moves at speedvbetween stops separated by a distanced.

Energy goes not only into the brakes: while the car is moving, it makes air swirl around. A car leaves behind it a
tube of swirling air, moving at a speed similar tov. Which of these two forms of energy is the bigger: kinetic energy
of the swirling air, or heat in the brakes? Let’s work it out.

• The car speeds up and slows down once in each durationdv. The rate at which energy pours into the brakes is:

kinetic energy

time between braking events

=

1

2 mcv

2

d

v

=

1

2 mcv

3

d

, (A. 1 )

wheremcis the mass of the car.

Figure A.3:A car moving at speedvcreates behind it a tube of swirling air; the cross-sectional area of the tube is
similar to the frontal area of the car, and the speed at which air in the tube swirls is roughlyv.

• The tube of air created in a timethas a volumeAvt, whereAis the cross-sectional area of the tube, which
  is similar to the area of the front view of the car. (For a streamlined car,Ais usually a little smaller than
  the frontal areaAcar, and the ratio of the tube’s effective cross-sectional area to the car area is called the drag
  coefficientcd. Throughout the following equations,Ameans the effective area of the car,cdAcar.) The tube
  has massmair=ρAvt(whereρis the density of air) and swirls at speedv, so its kinetic energy is:

I’m using this formula:

mass=density×volume

The symbolρ(Greek letter ’rho’) denotes the density.

1

2

mairv^2 =

1

2

ρAvtv^2 ,

and the rate of generation of kinetic energy in swirling air is:

1

2 ρAvtv

2

t

=

1

2

ρAv^3.